Questions: Polynomial and Rational Functions Finding the intercepts, asymptotes, domain, and range from the graph of ... The graph of a rational function f is shown below. Assume that all asymptotes and intercepts are shown and that the graph has no "hole Use the graph to complete the following. (a) Write the equations for all vertical and horizontal asymptotes. Enter the equations using the "and" button as necessary. Select "None" as necessary. Vertical asymptote(s): Horizontal asymptote(s): (b) Find all x-intercepts and y-intercepts. Check that apply. x-intercept(s): -4 -1 2 None y-intercept(s): -1 0 1 None (c) Find the domain and range of f. Write each answer as an interval or union of intervals. Domain: Range:

Polynomial and Rational Functions
Finding the intercepts, asymptotes, domain, and range from the graph of ...

The graph of a rational function f is shown below.
Assume that all asymptotes and intercepts are shown and that the graph has no "hole Use the graph to complete the following.
(a) Write the equations for all vertical and horizontal asymptotes. Enter the equations using the "and" button as necessary. Select "None" as necessary.

Vertical asymptote(s):

Horizontal asymptote(s):
(b) Find all x-intercepts and y-intercepts. Check that apply.
x-intercept(s):
-4
-1
2
None
y-intercept(s): -1
0
1
None
(c) Find the domain and range of f.

Write each answer as an interval or union of intervals.
Domain:
Range:

Transcript text: Polynomial and Pational Functions Finding the intercepts, asymptotes, domain, and range from the graph of ... The graph of a rational function $f$ is shown below. Assume that all asymptotes and intercepts are shown and that the graph has no "hole Use the graph to complete the following. (a) Write the equations for all vertical and horizontal asymptotes. Enter the equations using the "and" button as necessary. Select "None" as necessary. Vertical asymptote(s): $\square$ Horizontal asymptote(s): $\square$ (b) Find all $x$-intercepts and $y$-intercepts. Check $a / /$ that apply. $x$-intercept(s): $-4$ $-1$ 2 None $y$-intercept(s): $-1$ 0 1 None (c) Find the domain and range of $f$. Write each answer as an interval or union of intervals. Domain: $\square$ Range: $\square$

Solution

Solution Steps

Step 1: Identify Vertical and Horizontal Asymptotes

Vertical Asymptotes: These occur where the function approaches infinity. From the graph, the vertical asymptotes are at $ x = -1 $ and $ x = 1 $.
Horizontal Asymptote: This occurs where the function levels off as $ x $ approaches infinity. From the graph, the horizontal asymptote is at $ y = 0 $.

Step 2: Find x-intercepts and y-intercepts

x-intercepts: These are the points where the graph crosses the x-axis. From the graph, the x-intercepts are at $ x = -4 $ and $ x = 2 $.
y-intercepts: These are the points where the graph crosses the y-axis. From the graph, the y-intercept is at $ y = 0 $.

Step 3: Determine the Domain and Range

Domain: The domain is all the possible x-values for which the function is defined. From the graph, the function is not defined at $ x = -1 $ and $ x = 1 $. Therefore, the domain is $ (-\infty, -1) \cup (-1, 1) \cup (1, \infty) $.
Range: The range is all the possible y-values the function can take. From the graph, the function can take all y-values except $ y = 0 $. Therefore, the range is $ (-\infty, 0) \cup (0, \infty) $.